Lemma 2 (Maximal Inequality)  [1]} Let \(Z_1,\cdots ,Z_n \)  be real-valued random variables where a \(v>0\)  exists, such that for every \(i=1,2,\cdots ,n\) , we have \(\log \big (\operatornamewithlimits{\mathbb {E}}\big [\exp \left(\lambda Z_i\right)\big ]\big )\le \frac{\lambda v^2}{2}\) , then we have:
\(\operatornamewithlimits{\mathbb {E}}\left[\max _{i=1,2,\cdots ,n} Z_i\right] \le \sqrt{2v\log n}.\) 

